2.2.1 Simulataneous Confidence Sets: a bootstrap procedure

Here is presented a Bootstrap procedure to get a simultaneous confidece set for all the \(\rho_{i,j}\) of a random vector (Par.“High Dimensional Bootstrap For Pearson Correlation”) :

• Let \(\mathbf{R}\) be the \((D \times D)\) matrix of true correlations and let \(\mathbf{\hat{R}}\) be the \((D \times D)\) matrix of sample correlations.

• Now let \(\{\mathbf{X}_1^*,...,\mathbf{X}_n^*\}\) denote a bootstrap sample and let \(\mathbf{\hat{R}}^*\) be the \((D \times D)\) matrices of correlations from the bootstrap sample.

In this analysis: Very roughly speaking,it means that we’ve to build \(B\) (Bootstrap size) \(\mathbb{X}^*\) matrices by shuffling the rows of \(\mathbb{X}\) (\(n\) is the number of the rows, so the size of our \(iid\) sample for every stock) and then compute the associated \(\mathbf{\hat{R}}^*\).

• After taking this \(B\) bootstrap samples we have \(\{\mathbf{\hat{R}}^*_1,..., \mathbf{\hat{R}}^*_B\}\)

• Now, for each bootstrap sample \(b \in \{1, . . . ,B\}\), define the following bootstrapped replicate of a simultaneous test statistics \[\Delta_b = \sqrt{n}\,\text{max}|\mathbf{\hat{R}}^*_b[j,k] - \mathbf{\hat{R}}[j,k]|, \]

and its associated bootstrapped ECDF: \[ \hat{F}_n(t) = \frac{1}{B} \sum_{b=1}^{B} \mathbb{I}(\Delta_b \leq t)\]

• Within the usual bootstrap analogy, for large \(n\) and \(B\), \(\hat{F}_n(t)\) should be a good approximation to \[ F_n(t)=\mathbb{P}(\sqrt{n}\,\text{max}|\mathbf{\hat{R}}[j,k] - \mathbf{{R}}[j,k]| \leq t)\]

-Finally, to build our simultaneous confidence set, consider the sample quantile at level \(1 {-} \alpha\) of the bootstrapped distribution \(\hat{F}_n(t)\) , say \(t_\alpha = \hat{F}^{-1}_n(1- \alpha)\), and set \[ C_{j,k}(\alpha) = \Big[ \hat{R}[j,k]- \frac{t_\alpha}{\sqrt{n}} , \hat{R}[j,k]+ \frac{t_\alpha}{\sqrt{n}} \Big] \]

Theorem. If \(D=o(exp(n^{1/6}))\), then

\[ \displaystyle{\lim_{n \to \infty}}\mathbb{P}(\mathbf{R}[j,k] \in C_{j,k}(\alpha) \text{for all } (j,k)) = 1- \alpha \]

2.2.2 Bootstrapping the Global Economy (seriously?)

Now that we got the theoretical procedure to build our Pearson-based marginal correlation graph, it’s time to present some results. We implement the previous procedure in the following code and we obtain some interesting results. Just for consistence, these are the choosen parameters of the first simulation:

• \(D_{max} = 120\)
• \(\epsilon = 0.28\)
• Confidence sets at level \(\alpha = 0.05\) (95%)
• Bootstrap size \(B = 1000\)

The following chunk shows the creation of the boostrap replicates \(\{ \Delta_b\}_{b=1}^{b=B}\):

R_hat = cor(X) # Computes a plug-in estimate of the correlation matrix

D = dim(X)[2] # Gets the number of stocks
B = 1000 # Sets B

#This function gets X,R_hat and the number of entries, shuffles X
#rows in order to get the matrices of correlations from the bootstrap
#sample X_star and computes the statistic Delta.This procedure is
#still, clearly vectorized.

delta_generator <- function(x, df = X, R_h = R_hat, n = number_of_entries){
  idx_sample <- sample(1:dim(df)[1],dim(df)[1], replace = TRUE)
  X_star <- df[idx_sample,]
  R_hat_star <- cor(X_star)
  delta_b <- sqrt(n) * max(abs(R_hat_star - R_hat))
  return(delta_b)
}
# Here we get the resultsstored in a vector
res_d <- sapply(1:B, delta_generator) 

Now, just for completeness, it’s good to take a look to the empirical distribution \(\hat{F}_n(t)\) and density of the bootstrap replicates.

The next step is compute the quantile of interest from \(\hat{F}_n(t)\) and then the aforementioned simulataneous confidence set. At the end, we compute the adjacency matrix of the graph by testing the appartenance of any \(\rho(j,k)\) to it’s corresponding CS.

alpha <- 0.05 # Chooses alpha
t_alpha <- quantile(res, 1- alpha) # Compute the quantile of interest (res is the ECDF of Deltas)

#This function (again for a vectorized procedure) takes t_alpha, the
#number of entries,epsilon and returns the adjacency matrix of the
#graph by checking the intersection aforementioned.

adiacence_matrix <- function(x, t = t_alpha, epsilon = 0.28, n = number_of_entries){
  lower_bound <- max(-1,x - t/sqrt(n))
  upper_bound <- min(1,x + t/sqrt(n))
  if ((-epsilon > upper_bound)|(epsilon < lower_bound)){
    return(1)
  }
  else{
    return(0)
  }
}


#density <- function(eps){
A <- apply(R_hat,1:2, adiacence_matrix)

We have what we need to visualize the graph. For our first simulation we choose \(\epsilon = 0.3\) that, with the IID assumption, represents a pretty strong correlation, so we expect to visualize stocks clustered by sectors.

#This small function just allows us to assign color to nodes by
#their sector
sector_matcher <- function(x, df = Symbols_GISC){
  c_row <- Symbols_GISC[which(Symbols_GISC$Symbol == x),]
  return(c_row$Sector)
}

sectors <-sapply(colnames(A),sector_matcher)
ggnet2(A, color = sectors, palette = "Set3", node.size = 5, node.alpha = 7/10, edge.alpha = 7/10)

Nice! The cluseters are evident and we are pretty sure that stocks of the same sector are more correlated than stocks of different sectors. Later we’ll provide two measures of this “clustering”.

It’s interesting to observe the changes in the graph by varying \(\epsilon\).

In order to analyze this dependency we’ll show four kinds of measurements, two of these are the aforementioned measures of clustering intra-sector:

• A dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph. The distinction between sparse and dense graphs is rather vague, and depends on the context. For undirected simple graphs, the graph density is defined as the ratio between two times the number of edges over the total number of possible edges:

\[ \delta = \frac{| E|}{\frac{|V|(|V|-1)}{2}}\] where:

\(|V|\) is the cardinality of the set of vertices, so that denominator is the total number of possible edges.

\(|E|\) is the cardinality of the set of edges, so the number of edges.

In our case we can check how the density varies as a function of \(\epsilon\).

• A similiar concept ca be used to check that the stocks of the same sector are the most correlated and tend to cluster as \(\epsilon\) varies. A good measure can be: \[ \delta_c = \frac{ \sum_{i=0}^S e_i}{|E|} \] where:

– \(S\) is the number of sectors – \(e_i\) is the number of edges that link stocks of the \(i^{th}\) sector. – \(|E|\) is the cardinality of the set of edges, so the number of edges.

• Another measure can be defined as: \[ \delta_e = \frac{ \sum_{i=0}^S e_i}{\sum_{i=0}^S e_{max,i}} - \frac{P}{P_{max}} \] where:

– \(e_{max,i}\) is the number of all possible edges between stocks belonging to the \(i^{th}\) sector. – \(P\) is the number of edges that link stocks belonging to different sectors, so it can be written as \(|E| - \sum_{i=0}^S e_i\) – \(P_{max}\) is the number of all possible edges between stocks belonging to different sectors, so it can be written as \(\frac{|V|(|V|-1)}{2} - \sum_{i=0}^S e_{max,i}\)

It’s easy to observe that this quantity \(\in [-1,1]\) and it is \(1\) where there are perfect sectors clusters and \(-1\) when there are no links among sotcks of the same sector.

• A beautiful GIF that shows the evolution of the graph as \(\epsilon\) grows.

First of all, let’s take a look at the density:

#This function computes the density and saves the graphs as png in
#order to build our GIF :D

density <- function(eps, d = D){
A <- apply(R_hat,1:2, adiacence_matrix, epsilon =eps )
sectors <-sapply(colnames(A),sector_matcher)
plot=ggnet2(A, color = sectors, palette = "Set3", node.size = 5, node.alpha = 7/10, edge.alpha = 7/10)

#We used the next commented lines to build the gif
#ggsave(paste(as.character(eps),".png"), plot = last_plot(), device = "png", path =
#"C:\\Users\\Egon\\Desktop\\Universita\\SDS\\HW2")
#tiger <- image_read(paste(as.character(eps),".png"))
#tiger <- image_annotate(image = tiger, text = paste("Epsilon =", as.character(eps)),size =
#50,boxcolor = #'orchid', font = 'Trebuchet')
#image_write(tiger, path = paste(as.character(eps),".png"), format = "png")

return(((length(which(A!=0))-d)/2) / (d*(d-1)/2))
}

# Let's plot delta
epsx <- seq(0,0.7,0.05)
d <- sapply(epsx,density)
plot(x = epsx, y = d, col = viridis(2),lwd = 2.5, type = "l", xlab = TeX("$\\mathbf{\\epsilon}$"), ylab = TeX("$\\mathbf{\\delta}$"))

As we can see and as we can expect, the density of the graph decreases as \(\epsilon\) increases. No correlation survives on a threshold greater than about \(0.4\).

Now we’re going to show \(\delta_c\):

density <- function(eps, d = D){
A <- apply(R_hat,1:2, adiacence_matrix, epsilon =eps )

# Sectors of rows and columns
col_sectors <- sapply(colnames(A),sector_matcher)
row_sectors <- sapply(rownames(A),sector_matcher) 

es <- 0 # the numerator of delta_c
for(row_idx in 1:d){
  for(col_idx in 1:d){
    if(col_sectors[[col_idx]]==row_sectors[[row_idx]]){
      # if the sectors of an element of A are equals,add it to
      #the numerator
      es <- es + A[row_idx,col_idx] 
    }
  }  
}# Just our def of delta_c(-D becouse the aren't self-edges)
delta_c = (es-d)/(length(which(A!=0))-d)
if (is.na(delta_c)){
  delta_c = 0
}
return( delta_c)
}

# Let's plot delta_c
epsx <- seq(0,0.7,0.01)
d1 <- sapply(epsx,density)
plot(x = epsx, y = d1, col = viridis(2),lwd = 3.0, type = "l", xlab = TeX("$\\mathbf{\\epsilon}$"), ylab = TeX("$\\mathbf{\\delta_c}$"),panel.first=grid())

Also in this case the result is what we could expect. In fact,\(\delta_c\) grows as the threshold grows, due to the fact that only most correlated stocks survive and this stocks are the ones in the same sector. After, when epsilon is too high, the quantity goes down to zero.

The proof that in the maximum of the second curve there is the highest level of clustering by stocks of the same sector is that, at the same \(\epsilon\), in the first curve there is a very low density. Combining this observations, the only explanation possible is that the stocks of the same sector are the most correlated.

Now that we have a mathematical proof of the “high” correlation among stocks belonging to same clusters, we can use \(\delta_e\) to confirm this result.

density <- function(eps, d=D){
A <- apply(R_hat,1:2, adiacence_matrix, epsilon =eps )

# Sectors of rows and columns
col_sectors <- sapply(colnames(A),sector_matcher)
row_sectors <- sapply(rownames(A),sector_matcher) 

es <- 0 #Computing the numerator of the first addend
for(row_idx in 1:d){
  for(col_idx in 1:d){
    if(col_sectors[[col_idx]]==row_sectors[[row_idx]]){
      #if the sectors of an element of A are equals,add it to
      #the numerator
      es <- es + A[row_idx,col_idx] 
    }
  }  
}

B <- 0 #Computing the denominator of the first addend
for (el in GISC_cat){
  V_i = length(which(col_sectors==el))
  B <- B + V_i*(V_i - 1)/2
}# Just our def of delta_c(-D becouse the aren't self-edges)
return( ((es-d)/2)/B - ((length(which(A!=0))-d)/2-((es-d)/2))/((d*(d - 1)/2-B)))
}

# Let's plot delta_e
epsx <- seq(0,0.7,0.01)
d2 <- sapply(epsx,density)

plot(x = epsx, y = d2, col = viridis(2),lwd = 3.0, type = "l", xlab = TeX("$\\mathbf{\\epsilon}$"), ylab = TeX("$\\mathbf{\\delta_e}$"),panel.first=grid())

Last but not least, let’s check how the graph evolves increasing \(\epsilon\):

Is what we see for this realization the truth or it’s just a case we observed this measures? A good idea is searching for statistical significance.

2.2.3 Intra-sector Clustering: Hypothesis Testing

We wanna show that stocks belonging to same clusters are the most correlated. By definition of \(\delta\) and \(\delta_e\) and with the previous observations on their beahviour, we can give statistical significance to the “clustering” by demonstrating that this two indices are negative correlated for all the value of \(\epsilon\) that generate a graph with \(|E| >> 0\) (\(\epsilon \leq 0.5\)). So we can make a Wald test at level \(\alpha=0.05\) on this hypothesis:

\(H_0 : \rho(\delta,\delta_e) \ge 0\) and \(H_1 : \rho(\delta,\delta_e) < 0\)

Of course the value plotted in the previous paragraph are the sample we need. Let’s look what happens:

alpha = 0.05
cor.test(d1[1:50], d[1:50], alternative = "less", conf.level = 1 - alpha)

## 
##  Pearson's product-moment correlation
## 
## data:  d1[1:50] and d[1:50]
## t = -2.8684, df = 13, p-value = 0.006591
## alternative hypothesis: true correlation is less than 0
## 95 percent confidence interval:
##  -1.0000000 -0.2490237
## sample estimates:
##        cor 
## -0.6225757

As we can see, we are pretty sure to reject \(H_0\), and so can state that the last stocks to survive, the most correlated, are the ones that belongs to the same sectors.

2.4.1 Simple treshold checking

To build the graph using Distance Correlation we don’t have CI so a first, not simultaneous, approach can be:

• Compute all possible distance correlation among stocks.

• Check if their abs are greater than a threshold \(\epsilon\) and, in this case, place an edge between the corresponding stocks.

We implement this procedure in the following code, for \(\epsilon=0.27\)

alpha <- 0.1# Chooses alpha
D = 30 # A subset, due to computational issues
# This function returns the squared distance correlation
#for a pair of stock (still vectorized procedure)
dcor_dis <- function(x,y, df = X){ 
  rho <- dcor.test(df[x],df[y], R = 500)$statistic
  return(rho)
}
comb <- combn(names(X)[1:D], 2) 
D_hat <- mapply(dcor_dis, x = comb[1,],y = comb[2,]) # Returns a vector containing the
#squared distance correlations



edges_generator <- function(x,epsilon = 0.33){ 
  if (abs(D_hat[x])>epsilon){
    return(1)
  }
  else{
    return(0)
  }
}
# A vector of edges corresponding to the "correlated" pairs of stocks
edges <- sapply(1:length(D_hat),edges_generator) 

A1 <-matrix(NA, nrow = D, ncol = D) # Initializes the adjacency matris
row.names(A1) <- names(X)[1:D]
colnames(A1) <- names(X)[1:D]

for (idx in 1:length(edges)){ # And builds it
A1[comb[1,idx],comb[2,idx]] <- edges[idx]
A1[comb[2,idx],comb[1,idx]] <- edges[idx]
}
diag(A1) = 1

# As before
sectors1 <-sapply(colnames(A1),sector_matcher)
ggnet2(A1, color = sectors1, palette = "Set3", node.size = 5, node.alpha = 7/10, edge.alpha = 7/10)

It seems that this basic approach doen’t give us too much information, it’s a messy and not meaningful graph. We can go deeper.

2.4.2 Multiple Testing

To reach a simultaneous control on the probability of type 1 error of putting an edge if the correlation is under a choosen threshold \(\epsilon\) (the null hypothesis for the \((i,j)^{th}\) pair has to not being rejected), there are two ways:

• Family-Wise Error Rate: in this case, it’s requested

\[ FWER_\epsilon=\mathbb{P}(R_F(\epsilon) \ge 1) \leq \alpha\] where \(R_F(\epsilon)\) is the number of false positives for the threshold \(\epsilon\).

Bonferroni Correction to achive this type of error control.

• False Discovery Rate: a “relaxed” control on the mean of the ratio between false positive and total positive defined as

\[ \mathbb{E}\Big(\frac{R_F(\epsilon)}{R(\epsilon)} \Big)=\mathbb{E}(FDP_t) \leq \alpha \] and

\[FDR_t = \mathbb{E}(FDP_t) \]

Benjamini-Hochberg Correction to achive this type of error control.

2.4.3 Family-Wise Error Rate: Bonferroni Correction

This type of control,as seen above, is very strict and so it’s useful when false discoveries can’t be tolerated. Given a collection of hypothesis(\(m\) tests), the \(k^{th}\) null hypotheses is rejected if, for a given level of control \(\alpha\):

\[ q^{(k)} = min\{m \cdot p^{(k)},1\} \leq \alpha\] where \(p^{(k)}\) is the p_value of the \(k^{th}\) test. \(q^{(k)}\) is called Bonferroni q_value.

Let’s see how this strategy works for our purposes. The following code generates the graph associated to our data using Distance Correlation and Bonferroni Correction. Please note that formally we are testing \(m = {D_{sub}\choose2}\) hypothesis.

alpha <- 0.1 # Chooses 
#This function returns the p-values corresponding to distance correlation test
#for a pair of stock (still vectorized procedure)
dcor_dis <- function(x,y, df = X){ 
  rho <- dcor.test(df[x],df[y], index = 0.001, R = 500)$p.value
  return(rho)
}
comb <- combn(names(X)[1:D], 2) # All possible combination of stocks

# Returns a vector containing the p_values
D_hat_p <- mapply(dcor_dis, x = comb[1,],y = comb[2,]) 
edges_generator_p <- function(x,epsilon = 0.31, d = D){ 
  if (abs(D_hat_p[x])<= alpha/choose(d,2)){
    return(1)
  }
  else{
    return(0)
  }
}
# A vector of edges corresponding to the "correlated" pairs of stocks
edges <- sapply(1:length(D_hat_p),edges_generator_p)

A1 <-matrix(NA, nrow = D, ncol = D) # Initializes the adjacency matrix
row.names(A1) <- names(X)[1:D]
colnames(A1) <- names(X)[1:D]

for (idx in 1:length(edges)){ # And builds it
A1[comb[1,idx],comb[2,idx]] <- edges[idx]
A1[comb[2,idx],comb[1,idx]] <- edges[idx]
}
diag(A1) = 1

# As before
sectors1 <-sapply(colnames(A1),sector_matcher)
ggnet2(A1, color = sectors1, palette = "Set3", node.size = 5, node.alpha = 7/10, edge.alpha = 7/10)

From this test results that all the stocks are independent from the others. Actually, it’s not like this. The value of the p-values are not enough small and this is due to the small number of permutations \(R\) choosen. In fact, growing up \(R\) we can obtain, in this case, arbitrarily small p-values. After this observation we can state that all stocks are correlated. This makes sense, because distance correlation is able to detect also non linear dependencies. Anyways this strong correlation is still a bit weird. The explanation is that we’re assuming the observations are IID while they’re a time series and, under this hypothesis, we can count on a very large sample size \(n\) that allows to compute very high-power tests.

For these reasons p-values are small and are all equals (it’s not easy to estimate such small values). Moreover, this kind of distance is useful to check the dependency between two W-dimensional random vectors \(X\) and \(Y\) in the limit for \(W\) and \(n\). In this case \(n\) is big but \(W=1\) becouse every stock is an univariate r.v. . This makes the Type 1 Error control weaker and, in this sense, we can enforce our results by checking how the density of the graph \(\delta\) varies as \(\alpha\) goes down. In this way,we can figure out which is the critical value for \(\alpha\) (in light of what we’ve written) that swithces the graph from full connected to edge-less.

In the light of this observations we can try to use another kind of asymptotic test based on correlation distance that is implmented in R (in the package Rfast).This implementation has a very useful option that allows to return logged p-values, so the smallest values are dilated by \(log\). Let’s check if something changes by plotting the density of the graph as \(\alpha\) varies.

#This function returns the p-values corresponding to distance correlation test
#for a pair of stock (still vectorized procedure)
#detach("package::energy", unload=TRUE) # We have to switch the packages
library(Rfast)
D = dim(X)[2] # For this no permutation test we can use the whole dataset
dcor_dis <- function(x,y, df = X){
  rho <- Rfast::dcor.ttest(x = as.matrix(df[x]),y = as.matrix(df[y]), logged = T)
  return(rho[4])
}

comb <- combn(names(X)[1:D], 2) # All possible combination of stocks

# Returns a vector containing the p_values
D_hat_p <- mapply(dcor_dis, x = comb[1,],y = comb[2,]) 

edges_generator_p <- function(x, d = D,al){ 
  if (D_hat_p[x]<= log(al/choose(d,2))){
    return(1)
  }
  else{
    return(0)
  }
}

check_edges <- function(alpha){

# A vector of edges corresponding to the "correlated" pairs of stocks
edges <- sapply(1:length(D_hat_p),edges_generator_p, al = alpha)

A1 <-matrix(NA, nrow = D, ncol = D) # Initializes the adjacency matrix
row.names(A1) <- names(X)[1:D]
colnames(A1) <- names(X)[1:D]

for (idx in 1:length(edges)){ # And builds it
A1[comb[1,idx],comb[2,idx]] <- edges[idx]
A1[comb[2,idx],comb[1,idx]] <- edges[idx]
}
diag(A1) = 1

return(((length(which(A1!=0))-D)/2) / (D*(D-1)/2))

}

alphax <- seq(from = 0.01, to = 0.0001, by = -0.0005)
d_a <- sapply(X = alphax,FUN = check_edges)
plot(x = alphax, y = d_a, col = viridis(2),lwd = 3.0, type = "l", xlab = TeX("$\\mathbf{\\alpha}$"), ylab = TeX("$\\mathbf{\\delta}$"),panel.first=grid())

From the figure we can see that the graph is full connected also for very small values of \(\alpha\) and this is consistent with what we noticed above, the null hypothesis is always rejected. The positive thing, with this test, is that we can take a look to the distribution of the p-values to get a visual perception of what it’s happened. Moreover, we highlight the logged smallest tested \(\alpha/m\) and the logged high p-value to better understand that there is no way, with IID hypothesis on this kind of data, to not reject the null using this test procedure.

hist(D_hat_p, breaks = 80, col = viridis(1), prob = T, main = TeX("$\\Density\\,of\\,p-values$"), xlab = TeX("$p-values$"))

## [1] "Max log p-value: -2.17782209678423 Minimum log(alpha/m): -17.7572865215418"

2.4.4 False Discovery Rate: Benjamini-Hochberg Correction

It’s clear that what we’ve seen in the previous paragraph is still valid but, just for completeness, let’s try also the Benjamini-Hochberg Correction.

This is,generally (for meaningful cases), a weaker type of control than Bonferroni Correction. It allows us to control the mean of the aforementioned ratio between false positives and total positives. The following procedure achives this type of control:

• Find the ordered p_values: \(p^{(1)} < ... < p^{(m)}\)
• Let \(j = max\{k : p^{(k)} < k \cdot \frac{\alpha}{m}\}\)
• Decision rule: Reject all the hypothesis having \(p^{(k)} < p^{(j)}\)

Let’s see what happens with our data:

sorted_p <- sort(D_hat_p)
p_j = -1
idx = 1

# Finds p_j (they're all equal in this case,obviously)
for (el in sorted_p){
  if (el < idx * alpha/choose(D,2)){
    p_j = el
  }
  idx = idx + 1
} 
edges_generator_h <- function(x){ 
  if (abs(D_hat_p[x])<=p_j){
    return(1)
  }
  else{
    return(0)
  }
}
# A vector of edges corresponding to the "correlated" pairs of stocks

edges <- sapply(1:length(D_hat_p),edges_generator_h) 
A1 <-matrix(NA, nrow = D, ncol = D) # Initializes the adjacency matrix
row.names(A1) <- names(X)[1:D]
colnames(A1) <- names(X)[1:D]

for (idx in 1:length(edges)){ # And builds it
A1[comb[1,idx],comb[2,idx]] <- edges[idx]
A1[comb[2,idx],comb[1,idx]] <- edges[idx]
}
diag(A1) = 1

# As before
sectors1 <-sapply(colnames(A1),sector_matcher)
ggnet2(A1, color = sectors1, palette = "Set3", node.size = 5, node.alpha = 7/10, edge.alpha = 7/10)

Obviously ,for this applciations,this technique is useless too but we tought it would be useful to proof.

There are other ways to get some kind of more meaningful test for this situation, but this is not the place.